1、Basic Math for Process ControlBasic Math forProcess Controlby Bob ConnellNoticeThe information presented in this publication is for the general education of the reader. Because neither the author nor the publisher have any control over the use of the information by the reader, both the author and th
2、e publisher disclaim any and all liability of any kind arising out of such use. The reader is expected to exercise sound professional judgment in using any of the information presented in a particular application.Additionally, neither the author nor the publisher have investigated or considered the
3、affect of any patents on the ability of the reader to use any of the information in a particular application. The reader is responsible for reviewing any possible patents that may affect any particular use of the information presented.Any references to commercial products in the work are cited as ex
4、amples only. Neither the author nor the publisher endorse any referenced commercial product. Any trademarks or tradenames referenced belong to the respective owner of the mark or name. Neither the author nor the publisher make any representation regarding the availability of any referenced commercia
5、l product at any time. The manufacturers instructions on use of any commercial product must be followed at all times, even if in conflict with the information in this publication.Copyright 2003 ISA The Instrumentation, Systems, and Automation SocietyAll rights reserved. Printed in the United States
6、of America. 10 9 8 7 6 5 4 3 2ISBN 1-55617-813-1No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior writ-ten permission of the publisher.ISA67 Alexander Drive
7、P.O. Box 12277Research Triangle Park, NC 27709Library of Congress Cataloging-in-Publication Data Connell, Bob. Basic math for process control / by Bob Connell. p. cm. Includes bibliographical references and index. ISBN 1-55617-813-1 1. Chemical process control-Mathematics. I. Title. TP155.75 .C662 2
8、003 660.281-dc21 2002013091vContentsPREFACE ixChapter 1 TRIGONOMETRY AND CYCLIC FUNCTIONS 1Units of Measurement, 1Functions of Angles, 2Definitions, 3Quadrants, 3Frequency of Cycling, 5Sine of the Sum of Two Angles, 7Cosine of a Sum, 8Sine of a Difference Between Two Angles, 9Cosine of a Difference,
9、 10Potentially Useful Relationships, 10Chapter 2 DIFFERENTIAL CALCULUS 15Concept of Approaching a Limit, 16Procedure for Determining a Derivative, 19Chapter 3 INTEGRAL CALCULUS 29Problem Areas, 30Practical Uses of Integration, 31Integration Over a Specified Range, 39Table of Basic Integrals, 48Chapt
10、er 4 INFINITE SERIES 49Power Series, 50The nth Term, 50Test for Convergence, 51vi Table of ContentsMaclaurins Series, 54Taylors Series, 58Chapter 5 COMPLEX QUANTITIES 61Background, 61Graphical Representation, 63The Complex Variable, 64Trigonometric and Exponential Functions, 65Separating the Real an
11、d Imaginary Parts, 67Chapter 6 DIFFERENTIAL EQUATIONS 71Introduction, 71Philosophy, 72Definitions, 72Application, 74Differential Equations of the First Order and First Degree, 74Linear Differential Equations with Constant Coefficients, 80Second Order Linear Differential Equation with Constant Coeffi
12、cients, 80The Oscillatory Case, 83The Constant of Integration, 84Units, 89Partial Differential Equations, 91Chapter 7 LAPLACE TRANSFORMS 93History, 93Transforms of Derivatives, 95The Oscillatory Case, 103Chapter 8 FREQUENCY RESPONSE ANALYSIS 107Background, 107The Bode Diagram, 108Frequency Response
13、of a Time Constant Element, 109Frequency Response of a Dead Time Element, 109Combinations of Components, 111Period of Oscillation, 112Summary, 114Chapter 9 TRANSFER FUNCTIONS AND BLOCK DIAGRAMS 117Background, 117Transfer Functions, 118The Step Input Function, 118Time Constants, 119Dead Time, 120The
14、Value of the Transfer Function, 120Block Diagrams, 123Table of Contents viiConditions for Continuous Oscillation, 125The Transfer Function of a Closed Loop, 127Evaluating the Closed Loop Transfer Function, 127Chapter 10 THE ZN APPROXIMATION 131Historical, 131The ZN Approximation, 136Estimating the F
15、requency of Oscillation, 141Values for the Dead Time and Time Constant, 141Just How Good Is the Approximation?, 142Making a Process Reaction Curve, 143Chapter 11 UNITS, BEST VALUES, FORMULAS, AND OTHER GOOD STUFF 151True Value, 151Errors, 152Errors in Combinations of Quantities, 152Correction Factor
16、, 153Significant Figures, 153Conversion of Units, 155Converting Formulas to New Units, 157The Most Representative Value (MRV), 162Predicting Future Values, 165How Much Confidence in the Most Representative Value?, 167The Standard Deviation, 168Curve Fitting, 169INDEX 173ixPrefaceIn order to become p
17、roficient in any branch of technology, the knowledge required will have certain building blocks in its foundation. For students aspiring to become knowledgeable in process control, one of the important blocks is mathematics. As such, any student who is striving for the certifi-cation necessary to en
18、ter the process control field can expect to be sub-jected to one or more courses in mathematics.Unfortunately, courses in mathematics tend to be taught by instructors whose mathematical minds are far above those of their students. The same applies to the authors of the textbooks which are dutifully
19、purchased as an adjunct to the class room teaching. What this can mean then, is that courses in mathematics which are intended to lead to a knowledge of pro-cess control, can instead become an obstacle to success. The math course has to be passed, after all.When I was striving to comprehend control
20、theory, I had trouble with the math personally, not so much because it was to deep for me, but because of the way that it was presented. There were just too many gaps in the explanation. Consequently, in this text I have tried to present the mathe-matical concepts in the way that I wish they had bee
21、n laid out for me.In my own mind, I have an admiration and respect for mathematics, because mathematics is basically an exercise in thinking logically. Rules in mathematics are always hard and fast. From my personal observations of the way that many process control situations are dealt with in indus
22、try, it is unlikely that there is any branch of technology that is more in need of logical thinking.x Basic Math for Process ControlI confess, at the outset, that I am in no way an authority on mathematics. My knowledge of mathematics really doesnt go one centimetre beyond what this book covers. In
23、fact, if it were not for a wonderful stroke of luck in which I came into contact with Mrs. Florica Pascal, I would not have been able to complete this text. A superior mathematician in her own right, Mrs. Pascal reviewed chapters, corrected mistakes, and showed me how to solve problems which were be
24、yond my humble capabilities.All of this means that this text on mathematics was written not by an expert, but by an engineer who has to see, and understand, each step in the development of a mathematical entity. The way the text is written more or less bears this out. An accomplished mathematician w
25、ill likely find it trivial or boring. But to the student of process control who has to get through the math course which the curriculum requires, it may just prove to be helpful. And, if whatever help was provided carries on into the on-the-job phase, so much the better.Bob Connell11Trigonometry and
26、Cyclic FunctionsTrigonometry is a branch of mathematics concerned with functions that describe angles. Although knowledge of trigonometry is valuable in sur-veying and navigation, in control systems engineering its virtue lies in the fact that trigonometric functions can be used to describe the stat
27、us of objects that exhibit repeatable behavior. This includes the motion of the planets, pendulums, a mass suspended on a spring, and perhaps most rel-evant here, the oscillation of process variables under control.Units of MeasurementThe most common unit of measurement for angles is the degree, whic
28、h is 1/360 of a whole circle.A lesser used unit is the radian. Although the radian is not ordinarily used in angular measurement, it should be understood because when differen-tial equations, which occur in control systems engineering, are solved, the angles emerge in radians.On the circumference of
29、 a circle, if an arc equal in length to the radius of the circle is marked off, then the arc will subtend, at the center of the circle, an angle of 1 radian. The angle (or POB) in Figure 1-1, illustrates this.In line with this definition of a radian, the relationship between radians and degrees can
30、be worked out. The full circumference of the circle (length 2 r) subtends an angle of 360 at the center of the circle. An arc of length r will subtend an angle of degrees. r2r- 360180-=2 Basic Math for Process ControlTherefore 1 radian = deg, or radians = 180 .The actual value of a radian is 571745,
31、 although this value is hardly ever required in control systems analysis.If the base line OB in Figure 1-1 remains fixed and the radius OP is allowed to rotate counterclockwise around the center O, then the angle (or POB) increases. If the starting point for OP is coincident with OB, and OP rotates
32、one complete rotation (or cycle) until it is again coincident with OB, then the angle will be 360. From this it is evident that 1 cycle = 360 = 2 radians.Functions of AnglesLet be any acute angle for which OB is the base, and P be any point on the inclined side of the angle, as in Figure 1-2. A perp
33、endicular from P down to the base OB meets OB at point A.First, the ratio of any one of the three sides of the triangle POA, to either of the other sides, is a characteristic of the angle . In other words, if any of the ratios PA/OP, OA/OP, or PA/OA is known, then the angle can be determined from th
34、e appropriate tables.Note that the values of these three ratios do not depend on the position of P. As P moves out along the inclined side of the angle, OP increases, but PA and OA also increase in the same proportion. The values of the ratios depend on the size of the angle , but not on the locatio
35、n of P.Figure 1-1. A radian defined.OPrrB0180- Chapter 1 Trigonometry and Cyclic Functions 3DefinitionsIn the triangle POA, the ratio of the length of the side opposite the angle to the length of the hypotenuse, or PA/OP, is called the sine of the angle . The abbreviation sin is generally used, that
36、 is, PA/OP = sin .The ratio of the side adjacent to the angle to the hypotenuse, or OA/OP, is called the cosine of the angle . This is usually abbreviated cos, that is, OA/OP = cos .The ratio of the opposite side to the adjacent side, or PA/OA, is called the tangent of the angle . This is abbreviate
37、d tan, so that PA/OA = tan .There are, in addition, some other less common functions of the angle . These are defined as follows.QuadrantsThe complete circle is divided into four equal parts (called quadrants) by horizontal and vertical axes that intersect at the center of the circle. These Figure 1
38、-2. Functions of angles.OBP0AOPPA-1sin- cosecant (abbreviated cosec )=OPOA-1cos- secant (abbreviated sec )=OAPA-1tan- cotangent (abbreviated cot )=4 Basic Math for Process Controlquadrants are numbered 1 to 4, starting with the upper right quadrant and proceeding counterclockwise, as diagrammed in F
39、igure 1-3.When is an acute angle, the radius OP lies in the first quadrant. As the radius rotates counterclockwise and increases beyond 90, OP lies in the second quadrant. For between 180 and 270, OP is in the third quadrant. For between 270 and 360, OP is in the fourth quadrant.By definition, the m
40、easurement of the radius OP is always positive. The measurement OA is defined to be positive when the point A is on the right side of the vertical axis, and negative when A is on the left side of the ver-tical axis. The measurement PA is defined to be positive when P is above the horizontal axis, an
41、d negative when P is below the horizontal axis.This means that sin , cos , and tan can have positive or negative values depending on the quadrant in which OP lies, which in turn depends on the magnitude of the angle . The following values consequently prevail.When = 0, PA = 0, and OA = OP. Therefore
42、 sin = 0, cos = 1, and tan = 0.When = 90, PA = OP, and OA = 0. Therefore sin = 1, cos = 0, and tan becomes infinite.When = 180, PA = 0, and OA = OP in magnitude, but OA is negative and consequently OA/OP = 1. Therefore sin = 0, cos = 1, and tan = 0.Figure 1-3. The quadrants.OPB0123 4AChapter 1 Trigo
43、nometry and Cyclic Functions 5When = 270, PA = OP in magnitude, but PA is negative, so PA/OP = 1. Therefore, sin = 1, cos = 0, and tan becomes infinite in the negative direction. The following table summarizes these points.The values in the table above show that the sine, cosine, and tangent func-ti
44、ons repeat themselves with time and are therefore cyclic. Periodic is another term that is sometimes used. The sine and cosine functions both cycle between +1 and 1, while the tangent function cycles between + and .Therefore, sine and cosine functions are useful in describing the behavior of objects
45、 and systems that are cyclic. As an example, an object might be known to cycle between the limits of 0 and 10. Its behavior y could be described using the sine function, asy = 5 sin + 5 = 5 (sin + 1).In this relationship, when sin = 1, y = 10, and when sin = 1, y = 0.Frequency of CyclingIf the motio
46、n of an object is linear, the distance traveled is equal to the average velocity of the object multiplied by the elapsed time. In symbol form,s = v twhere s is in metres, v is in metres per second, and t is in seconds.The equivalent relation for rotational motion, as in the case of the radius rotati
47、ng around the center of its circle, is that the angle swept through is equal to its angular velocity multiplied by the elapsed time, or in symbol form, = twhere is in radians, (the Greek letter often used for angular velocity) is in radians per second, and t is in seconds.Table 1-1. Sine, Cosine and
48、 Tangent FunctionsQuadrant sin cos tan 1 0 to 90 0 to 1 1 to 0 0 to 2 90 to 180 1 to 0 0 to 1 to 03 180 to 270 0 to 1 1 to 0 0 to 4 270 to 360 (= 0) 1 to 0 0 to 1 to 06 Basic Math for Process ControlThus, the function sin t rather than sin may be used to describe the behavior of objects and systems that cycle on a time basis. These include the motion of the planets, pendulums, masses suspended on springs, and the variation with time of temperature, p, and other plant variables that are being controlled.It is a rule of mathematics that the argument of a sine or cosine function does not have
